Enhancing teachers' noticing around mathematics teaching practices through video-based lesson study with peer coaching

Abstract

Purpose

The purpose of this study is to examine what teachers notice in their own enactment of eight high leverage practices as well as the patterns of interactions between the teachers and their peers when participating in video-based lesson study.

Design/methodology/approach

Each teacher taught and uploaded video from one lesson to a platform, which allowed video annotation, for their lesson study team. There were nine lesson study teams. This study used a qualitative design to examine the teachers' comments on their own videos as well as the patterns in the comments between peers on lesson study teams.

Findings

Teachers noticed both positive instantiations as well as opportunities for growth in their enactment of: using and connecting mathematical representations, posing purposeful questions and supporting students' productive struggle. Analysis displayed a pattern of exchanges where peers coached, validated, empathized and pushed each other beyond their comfort zone as critical peers.

Research limitations/implications

Although not all lesson study teams were made up of school-based teams and the teachers shared short recordings of their teaching, this research contributes to the understanding of how adapting lesson study by using video can help teachers notice their instantiation of teaching practices and peers can support and push one another towards ambitious instruction. Future research could extend this work by investigating the impact of video-based lesson study on teachers in isolated areas who may not have professional learning networks.

Practical implications

Video-based LS may help to overcome barriers to the implementation of lesson study, such as the challenge of scheduling a common release time for lesson observation and the financial burden of funding substitute teachers for release time.

Originality/value

The current realities of COVID-19 creates an opportunity for mathematics educators to reimagine teacher professional development (PD) in ways that push the field forward. In light of this disruption, the authors propose an innovative model of utilizing video-based Lesson Study (LS; Lewis, 2002) with peer coaching to offer PD opportunities with methodological considerations for both mathematics researchers and teacher practitioners. The authors document and analyze a collection of online LSs that were taught by a focal teacher and recorded for the peers in the LS group. Video-based LS PD structure allowed the authors to examine how they can leverage this online model of LS to analyze student thinking and learn about teaching rich tasks in an online environment using eight teaching practices. Through their paper the authors will detail the necessary features of online LS specifically using a video annotation tool like Goreact and how video can be used to enhance the professional learning of the mathematics teaching practices (MTPs; NCTM, 2014) and the noticing of student thinking (Jacobs et al., 2010; Sherin and van Es, 2009; van Es and Sherin, 2002, 2008). In addition, the authors will document the norms that were established in the online LS community that impacted collaboration of LS teams and developed strong peer coaching relationships. The online LS PD design also supports collaboration of teachers from varying contexts, promotes professional growth and demonstrates how educators might leverage peer coaches as social capital within their schools to develop teachers along the professional continuum.

Citation

Suh, J., Gallagher, M.A., Capen, L. and Birkhead, S. (2021), "Enhancing teachers' noticing around mathematics teaching practices through video-based lesson study with peer coaching", International Journal for Lesson and Learning Studies, Vol. 10 No. 2, pp. 150-167. https://doi.org/10.1108/IJLLS-09-2020-0073

Lesson study (LS) (Lewis, 2002) has long been an effective tool used to support teacher development by fostering pedagogical knowledge through iterative cycles of planning, teaching and assessing research lessons (Lewis et al., 2004) and has been embraced around the world for several decades. The appeal of Lesson Study was eloquently stated in the recent handbook, Theory and Practice of Lesson Study in Mathematics, by Huang et al. (2019) stating, “A key benefit of lesson study for teachers is that it moves much of the hard and complex work of teaching outside the classroom, expanding the time and space available to work on improving practice” (p. vi). The structure of LS allows teachers to engage deeply with the content they are to teach, to learn from their peers and a knowledgeable other, and to reflect on and improve their practice (Takahashi and McDougal, 2016). Although LS has been embraced in the US, there are some challenges encountered due to the nature of school schedules and the additional resources it requires in the US context (Suh et al., 2017). US-based researchers and mathematics educators have typically required grant-funding to pay for substitute teachers in order to provide teachers release time to engage in the rich planning, observing, and debriefing that is inherent to LS. Video-based LS provides an opportunity to resolve some of these challenges of teachers needing to be available at the same time to plan, observe, and debrief together. This study describes how teachers engaged in a hybrid, face to face and online collaboration for planning, and used video-based LS to reflect on their own enactment of eight high leverage teaching practices (NCTM, 2014) and how the LS teams interacted to support one another's development of these practices.

The current realities of COVID-19 create an opportunity for mathematics educators to reimagine teacher professional development (PD) in ways that push the field forward. In light of this disruption, we propose an innovative model utilizing video-based Lesson Study with peer coaching to offer PD opportunities with methodological considerations for both mathematics researchers and teacher practitioners. The purpose of this study is to examine what teachers notice in their own enactment of eight high leverage practices recommended by the National Council of Teachers of Mathematics (NCTM, 2014) as well as the patterns of interactions between the teachers and their peers when participating in video-based lesson study.

Conceptual framework

Adaptation of lesson study to educational contexts around the world

The global reach of Lesson Study demonstrates the flexibility of this elegant teacher-led professional development model. In fact, the ways Lesson Study has been adopted and modified in different countries is presented annually at the World Association of Lesson Study (WALS) conference. Through these international forums, we as teacher educators learn more about different educational cultures and policy contexts that explain the variations in how LS is adopted and adapted. Norwich (2018) identified five variations in Lesson Study practices which included (1) Learning study, a hybrid between lesson study and design experiments (Kullberg, 2010; Cheung and Wong, 2014); (2) Collaborative lesson research, a clear research purpose, study of available curricula, written research proposals, live research lessons and discussion, knowledgeable others and sharing of results (Takahashi and McDougal, 2016); (3) Lesson Study for Learning Community, which focuses on teacher development underpinned by a social justice commitment (Saito and Atencio, 2015); (4) Inter-professional lesson study, focused on adapting teaching for students with learning difficulties through inter-professional collaboration between teachers and support professionals (Norwich et al., 2018); (5) Micro-teaching lesson study, utilizing role play, reflection and debriefing (Regan et al., 2016).

Many of these adaptations maintain the integrity of Lesson Study, focusing on collaborative design of lessons or units of study; execution of the design with observation; and reflection on the product with a view towards improvement (Ono et al., 2013). In addition, some of these variations in the United States take advantage of different personnel or structures such as instructional coaching and data teams and professional learning communities as well as leverage the partnership between school districts and a university through grants providing extended content-based seminars with Lesson Study as follow-ups (Suh et al., 2017; Nickerson et al., 2014).

While acknowledging the need for variations and adaptations of Lesson Study across different contexts, Takahashi and McDougal (2016) cited five characteristics that are essential to the effective use of Lesson Study, including (1) a focus on the refinement of pedagogical expertise through a clearly defined research purpose; (2) a school- or district-wide initiative, not individual; (3) time for “kyouzai kenkyuu” (the study of teaching materials) is essential; (4) Lesson Study cycle, with live observation and detailed post-lesson evaluation, is conducted over several weeks, not in a few hours; and, (5) knowledgeable others who play a very important part at all points in the Lesson Study process and the post-lesson evaluation.

Supporting teacher noticing and teaching practices through video

Teachers, as with other professionals, develop the ability to notice opportunities to act as they gain expertise in their field (Mason, 2002). One way in which teachers' noticing skills have been developed is through video clubs (Sherin and van Es, 2009). The advancement in video reflection tools allows users to not only mark, but annotate specific instances to highlight, represent, and decompose practice (Jacobs et al., 2010; Sherin and van Es, 2009). Studies show that teachers who examine classroom video learn to focus on the mathematically important details of student thinking (Goldsmith and Seago, 2011; van Es and Sherin, 2002, 2008; van Es, 2011). We posit that embedding video within Lesson Study has the added benefit of “expanding the time and space” (Huang et al., 2019, p. vi) to support teacher noticing by slowing down and breaking down the complexity of teaching and learning.

Effective teaching in mathematics supports students' conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (National Research Council, 2001). The US's National Council of Teachers of Mathematics (NCTM, 2014) put together a set of eight research-based teaching practices that support students' learning in mathematics (see Table 1). These practices are referred to as the Mathematics Teaching Practices (MTPs). Becoming an effective teacher of mathematics necessitates being able to use not just one or a few of these practices, but all of them. Lesson Study aimed at supporting teachers' noticing of these practices in both their own and their peers' instruction, may provide the opportunities that teachers need to reflect on and improve their practice.

Video is a powerful tool for inquiry-based professional learning in practice-based settings (Borko, 2016) because it gives teachers the unique opportunity to view their own classroom practice, reflect on what they see, and engage in evidence-based discussions that can lead to changes and improvements in their practice. Enacting ambitious mathematics instruction, specifically supporting teacher noticing and use of discursive practices, can be challenging because of the complexities of teaching as well as the challenge of finding inservice teachers who are modeling ambitious mathematics instruction in the classrooms in which novice teachers can observe (Lampert et al., 2010). The use of video technology in teacher preparation and PD can enhance teachers' practice. Video artifacts have affordances in allowing a classroom teacher to highlight aspects of classroom life that they might not notice in the midst of carrying out a lesson. Borko et al. (2017) found that teacher leaders' facilitation of discussions during workshops was dependent on the purposeful selection of appropriate and “rich” video clips to foster productive discussions. While teaching certainly involves making judgments about what went well or poorly in a lesson, it is critical for teachers to first attend to what is significant in a classroom interaction, interpret that event and then use those interpretations to inform their pedagogical decisions (Jacobs et al., 2010; van Es and Sherin, 2008).

Research on noticing has been shown as a key ingredient to professional learning. The Association of Mathematics Teacher Educators (AMTEs) define noticing as one of the attributes of well-prepared teachers, “Well-prepared beginners commit themselves to noticing, eliciting, and using student thinking to assess [emphasis added] student progress in understanding the mathematics and to adjust instruction…” (AMTE, 2017, p. 2). Noticing can take place individually as well as collectively. Collaborative viewing and analysis of the video has been a component of several effective PD experiences, helping teachers to notice more specific teaching practices (Sherin and van Es, 2009; van Es and Sherin, 2008) and develop their professional vision (Sherin and van Es, 2009). This collaborative viewing has the potential to further “offload the challenging intellectual work to outside the classroom, when teachers have time to think through their instructional problems with other teachers who share the same problems” (Huang et al., 2019, p. vi) which is an affordance of lesson study.

Peer coaching

In some school districts in the United States, mathematics teacher leaders are identified as mathematics coaches or specialists. A number of research studies describe positive changes in teachers' practice as a result of interacting with an elementary mathematics specialist professional, including actively engaging students, emphasizing reasoning and problem solving over skills-based lessons, using students' work to inform instruction, and effectively planning lessons (Campbell, 1996; McGatha, 2009; Suh et al., 2017). Despite the position statement and recommendation for the use of Elementary Mathematics Specialists(EMSs) set forth by the Association of Mathematics Teacher Educators in the US in pre-K–6 environments (AMTE, 2013), there is the reality and challenge of staffing an EMS or coach in every school. Kraft et al. (2018) describe the challenges in building a corps of capable coaches whose expertise is well matched to the diverse needs of teachers in a school or district. They discussed a common approach to filling the demand for high-quality coaches and providing an alternative solution: to tap expert local teachers.

Bridging the research on effective Lesson Study and the benefits of teacher noticings through video studies, our project combined these two effective structures to examine how video-based peer coaching within online Lesson Study might support teachers' professional growth through the sharing of their professional noticing practices. We analyzed how teachers and peer coaches provided feedback and shared their professional noticing while they extended their teaching repertoire by taking on NCTM's (2014) eight MTPs. Our research was guided by two research questions:

1. What do teachers notice about their own enactment of the Mathematics Teaching Practices (MTPs) when participating in video-based LS?

2. What patterns of interaction existed among the peer coaches and focal teachers?

Methods

Context and design of the professional development

The design of the summer/fall PD institute illustrated in Figure 1 attends to the five core elements of Lesson Study (Takahashi and McDougal, 2016) depicted in the center, and describes the targeted ways we adapted video-based LS to attend to each of these core elements of high-quality, research-based PD (Desimone et al., 2002).

First, we recruited school teams of teachers and coaches, where two or more teachers constituted a team to encourage collective participation. We recruited school teams in order to sustain teacher learning as well as align with district and school-based initiatives.

The PD was launched with a face to face five-day summer institute consisting of 30 h with another 15 h of follow up sessions online in the fall to study teaching materials “kyouzai-kenkyuu” and to plan a lesson to enact and present the result of the research lessons at the Fall Lesson Symposium. Throughout the summer institute, we focused on the refinement of pedagogical expertise by focusing on the research-based eight MTPs endorsed by the National Council of Teachers of Mathematics document called Principles to Action (NCTM, 2014). Following the summer institute, the PD instructors grouped teachers by school and grade level into LS groups with three to five people in each group. Each teacher chose for themselves one of the eight MTPs as their research goal. Then each LS group created several rich mathematics tasks together, allowing the teacher to choose one task to teach, record and share.

Each teacher implemented their task in their classroom, video recorded their instruction, and posted a purposeful selection of 5–7 min video clip of that lesson to the video platform (see Figure 2). Members of the LS group all shared their videos with one another and were asked to use the eight MTPs as a framework for watching one another's videos, focusing primarily on the MTP that the focal teacher had identified as their personal goal. The two instructors of the course served as “knowledgeable others” but relied more on the group members to serve as peer coaches, and in some cases, serving as a “knowledgeable other” by providing peers with both strengths and areas for growth (i.e., “glows and grows”) related to the practices, without being evaluative. These guidelines were meant to foster a collegial environment wherein peers could validate one another and also offer useful critiques. The videos were intended to inform revisions to the tasks before being implemented in the next teacher's class.

Participants

All 32 teachers and six coaches who participated in the PD were invited to participate in this study. Only two teachers denied consent leaving us with 30 teachers (including 18 elementary, 11 middle school, and 1 high school) and the six coaches. The mean years of experience for these teachers and coaches was 12 years with the least experienced being 1 year to the most being 35 years. Six of the nine LS teams had at least one other colleague from the same school. Unfortunately, we do not have data on these teachers' prior experiences with LS or technology. However, anecdotally, many teachers expressed that LS and videoing their own instruction was a new experience for them.

Data sources

The data sources that we analyzed included video clips of the focus lessons using the GoReact software, which allows for participants to view, mark and annotate with timestamps linking comments to the instances. The focal teacher's annotations of the lesson on the video platform, and the exchanges among peer coaches on the video platform were downloaded for analysis for each group. There were a total of nine lesson studies that were conducted through the PD with groups of teachers and coaches in teams of four or five.

Nine video LS peer coaching groups' exchanges were analyzed using Dedoose, a qualitative analytic software. Each group's peer exchanges included the focal teacher's annotated reflections as well as the peer coaches' comments on glows and grows. Each group collaboratively planned a unit on a specific topic but taught and videotaped different lessons within the unit plan to share with their peer coaches. The interesting dynamic in the peer coaching groups was that there was not any hierarchy in that each of them served as a coach and a coachee as they switched roles when they shared their individual lessons.

Data analysis

In order to answer the first research question, we engaged in multiple rounds of data analysis. We first read and reread the peer exchanges in their entirety in order to get a sense of the data. As we read through the data, focusing on teachers' noticing of their own enactment, we observed that comments on their own videos seemed to express either a positive instantiation of a practice or a critique of their implementation of a practice. Thus, we decided in the next round of data analysis to apply a priori codes (Saldaña, 2009) from each of the following categories: (1) the eight MTPs and (2) “positive instantiation” or “self-critique.” We decided that the unit of analysis for each code should be one entire comment made by one teacher with regard to their own teaching practice (i.e., we excluded comments from peers as well as teachers' comments about their own students or other things such as their makeup). We also decided that each comment could be coded as both positive and self-critique, as well as being coded with multiple MTPs, if the teacher brought all these ideas into one comment. For example, “I felt like my questioning was good and definitely helped deepen their understanding from concrete to abstract [coded: positive instantiation and posing purposeful questions]. I felt like I guided them with the manipulatives too much possibly [coded: self-critique and productive struggle].” In this comment the teacher noted something she did well (i.e. question) and something she felt she needed to work on (i.e. guiding too much). Thus, the total number of codes exceeds the total number of comments coded. In the last round of data analysis for the first research question, we examined the co-occurrence of “positive instantiation” and each MTP as well as “self-critique” and each MTP to see if there were patterns in what the teachers noticed about their enactment of the MTPs.

With regard to the second research question, we conducted an intensive discourse analysis (Gee, 2014) to analyze the nature of the peer exchanges. Gee's Building Tasks Framework (2014) provides a framework for understanding what people are doing with their discourse. It also helps explain how people's use of language reveals their social identities and their theories regarding how the world works, as people use language to accomplish certain social goals. More specifically, according to Gee, people use language to build particular social identities, relationships and significance of meaning. Using Gee's (2014) approach and based on the initial analysis, we examined ways in which the focal teacher who taught the lesson and the peer coaches engaged in discourse around their teaching practices, attending to the use of language to bring significance to three building tasks: meaning, connections, and relationships. For each building task, we analyzed how members used the teaching practices “social languages,” and how they ascribed importance to the instances marked on the video and their critical reflection, and finally how they connected with other members by building on conversations and practices of one another. See Appendix 1 for the codebook.

Findings

Teacher noticings about their enactment of the MTPs

For the first research question regarding what teachers noticed about their own enactment of the MTPs, we found that teachers instantiated their own practice with both positive comments as well as self-critique (i.e. noting where they missed an opportunity or noticed a practice that they wanted to improve upon in the future). Since each focal teacher had chosen an MTP as their personal goal, teachers marked “instances where they intentionally enacted that practice” when annotating the video clip shared with the LS groups. Among their noticings, our data showed the instantiations of the following MTPs most often (see Table 2): (1) use and connect mathematical representations, (2) pose purposeful questions and (3) support productive struggle in learning mathematics. Within each of these practices we noticed that the teachers offered more self-critiques as compared to positive instances for posing purposeful questions and supporting productive struggle, whereas using and connecting mathematical representations was more even.

Highlighting moments of using and connecting mathematical representations

Annotations about using and connecting mathematical representations had high frequency in the video clips because it usually became the focus of the clips that teachers chose to share with their LS teams. About half of these annotations were positive as teachers shared successes and acknowledged efforts in growing in their designated practice. For example, in the following excerpt, the teacher (all names are pseudonyms) pointed out how she paid attention to students' use of mathematics representations to connect big ideas and connect concepts to procedural understanding.

Since my MTP was “using and connecting mathematical representations,” this conversation helped them advance their understanding of concepts and procedures while being able to make a choice about how they would represent their number. Them practicing the physical representation of the number would allow them to deepen their understanding when creating a visual which would then help them contextualize mathematics ideas.

In this excerpt, the teacher did not just note that she had used mathematical representations, but specifically reflected on how this MTP allowed students to “deepen their understandings.”

Struggles with using and connecting mathematical representations usually related to teachers wishing they had provided multiple representations. As this teacher shared, she wished she had helped students connect multiplication with a visual array.

The student only used an algorithm, They showed that they knew multiplication was repeated addition and could also be represented by an array, but I should have pushed further for one student to tell me the array.

Again this excerpt shows a teacher not only reflecting on her own instruction as she watched the video, but considering her implementation of the MTP in relation to the student thinking evident in the video.

The challenge of posing purposeful questions

As with using multiple representations, teachers also noted positives and self-critiques for posing purposeful questions. However, with regard to this MTP they noticed more than twice as many opportunities for improvement as positive instantiations. In a positive noticing of her own enactment, Rebecca noticed that she often asked, “What should we do?” throughout her lesson and acknowledged the struggle with avoiding asking funneling questions:

“What should we do?” is an example of questions that filled my lesson. My goal was to elicit and use evidence of student thinking. I believe, for the most part, I achieved this goal. When I found out that they were on the path of thinking about ratios (through questioning) as a strategy, we went for it! I tried very hard to avoid funneling questions.

The teacher was explicitly using purposeful questioning and differentiated between focusing and funneling patterns of questioning (Herbel-Eisenmann and Breyfogle, 2005; Wood, 1998) moving away from getting students to the right answer by asking a series of low-level, closed questions. She wanted her questioning to be more about focusing and responding to students' ideas. By posing this question to the group she was inviting specific reflection from others – an interaction that could not have occurred if the teachers were not engaging in LS groups.

At times, teachers were hard on themselves in their self-reflection as they shared what they noticed in their teaching video clip. Oftentimes, they critiqued how they responded in the moment but also offered ways they considered improving their practice. Jessica annotated on her video about the challenge she felt posing question that focused on student thinking,

I have such a hard time listening to students/understanding exactly what they're doing that I do not always use their thinking to pose questions to move them along or to pull their thinking out.

This excerpt represents how Jessica noted her area of weakness in the teaching practice with a self-critique but also pushed herself to improve on this practice by finding a strategy that helped her better understand students' thinking with this follow-up annotation,

When doing a task with another class to help me understand students thinking, I had them explain to me, but that just wasn't enough so I started writing down the math as they explained it to me- asking the student “like this…” then if I did not represent their idea accurately they continued to explain until I understood their strategy. Although I cannot do this with every single student, it was nice to see with a few students.

Here she shares how she is working on being more responsive to student thinking by having the opportunity to teach the same lesson in another block as well as scribing when students explain their thinking so she can follow their thoughts. Teachers' noticing their own shortcomings as well as ways they hoped to improve their practice in the future seems promising for changing future practice.

Teachers noticed areas of growth for supporting students' productive struggle

The pattern of teachers' noticing of their own enactment of productive struggle was similar to their noticing of purposeful questions: nearly twice as many instances of self-critique as compared to positives. In a positive noticing of her enactment of productive struggle, Cindy marked a segment where a student showed confusion and instead of rescuing them, she provided more time and space along with prompts for thinking. “I see some confusion here over even and odd. I prompt for more thinking, encouraging mathematical discourse and even some productive struggle.” This comment indicates that Cindy was able to notice the student's productive struggle and how she supported them.

Oftentimes, the missed opportunity the teachers noticed in their enactment of this MTP related to “rescuing students too quickly” instead of allowing students to experience productive struggle. For instance, Brenda said “I think I could have asked her to explain her strategy instead of me telling her strategy. I tend to rescue students and do not let them struggle to find the answer on their own.” In fact, Veronica stated it best about the struggle to promote productive struggle,

One of the biggest struggles I have when questioning students is to not allow them the opportunity to struggle to work through it. I tend to want to offer them too much information to guide them through a problem versus allowing them to work through it, particularly when a student has no clearly defined strategy they are using.

These missed opportunities were telling in that it demonstrated that teachers were not only celebrating their instantiation of the practices, in fact, they were just as aware of the moments in their teaching where they missed the opportunity for the instantiation of their practice.

Patterns of interaction among peer coaches and focal teachers

To address the second research question related to the opportunities afforded through video-based LS in terms of collegial interaction across peers, we analyzed the patterns of interaction that existed in the peer exchanges. Across the nine video LS groups, we found that the exchanges focused on the MTPs fell along four recurring patterns of peer coach interactions: (1) complimenting a teacher's move, (2) pressing teachers toward more ambitious teaching, (3) validating a common challenge, and (4) connecting to student thinking (see Table 3).

It was evident by the table that peer coaches complimented the teachers more often than press the teacher and question their practice. In fact, there were several instances, where peer coaches validated a common challenge and share how they also struggle with a practice. Peer coaches commented most about what they noticed about student thinking. Rather than present representative quotes for each of these themes disconnected from the discourse of the group, we present an entire peer exchange from one LS group here in order to showcase how the group members interacted to coach the focal teacher (see Table 4). These themes are defined with examples in the codebook in Appendix 1

We present data from the example lesson study group in Table 4. The first column provides a brief excerpt of the teacher and student discourse in the LS video, the second column illustrates the focal and peer coach exchanges that were annotated on the video, and the third column illustrates the patterns of interaction in their comments. To put this excerpt in context, below Table 4, we describe the task used in the examples lesson study exchange. Next we describe the video itself. Last, we provide a description of what the focal teacher and the peers noticed in the video, how they made connections to the teaching practices, as well as how they used language to support one another and build on their professional relationship.

Example lesson study group exchange: mathematics task description

One of the teachers recorded a snapshot of her interactions with a group of students based on an activity entitled the Crab Walk Relay Race (see Appendix 2) which focused on building three teams of approximately equal speed based on a given data set of times with two decimal places. This task requires students to use number sense to estimate teams with equal times (students can estimate the sums or they can reason about times that are approximately equal and distribute these evenly amongst the teams), to add decimal numbers in order to determine total times for each team, and to compare decimals using < , > , and =.

Example lesson study group exchange: video description

The video begins with the teacher (Vanessa) eliciting student explanations for how they went about solving the problem. While interacting with the students, the teacher personalized and empowered the students by asking them which team they would rather be on. This was a powerful way to engage students' errors by giving them some interest in the outcome. Rather than point out the inequities of the teams or other calculation errors, Vanessa simply asked them about what team they would want to be on. The teacher engaged in strong self-reflection in her written annotation about what happened in her classroom. In one instance, Vanessa reflected that she did not handle a particular student's description of her work as thoughtfully as she would have liked.

Example lesson study group exchange: patterns of interaction

In her video annotations Vanessa noticed a positive instantiation of MTP 8 and critiqued her own enactment of MTPs 5, 8, and 4. She marked the segment in her video where she missed an opportunity to make the mathematics more visible and critiqued that she did not further probe the student's thinking. These codes relate to our first research question. In the video, Vanessa showcased several strategies students used including using rounding as a technique to try to put together the crab walk teams while others ranked the students from slowest to fastest and marked the outliers or found the average to balance the teams. This was where the peer coach made an important connection to student thinking and complimented the focal teacher about eliciting a variety of student strategies. In the video, Vanessa then moved from the individual student to the group, soliciting student feedback as well. In support of her enactment of her goal to facilitate meaningful discourse, her peer, Dana, inquired if there would have been value in having the other students at the table listen to the first student's approach before discussing their own approach thereby bringing significance to the situation of comparing multiple representations as a way to deepen student thinking as well as elicit diverse strategies. In this way, Dana was pressing Vanessa toward more ambitious instruction. The knowledgeable other, Katie, continued to press the focal teacher to encourage more student-to-student discourse by bringing significance to Dana's suggestion. In this exchange, the peer coach and knowledgeable other complimented the teacher, pressed her to think about next steps, and connected their noticings to student thinking. They suggested to Vanessa that she consider working on ways to orient students to each other’s' thinking. This peer exchange shows how the video annotation platform allowed for self-reflection as well as collaborative group interactions and showcases the types of interactions that occurred across all the LS groups.

Discussion

We documented and analyzed a collection of online LS self-reflections and peer exchanges. The video-based LS structure allowed us to examine how we can leverage this video-based model of LS to focus teachers' reflection on students' mathematical thinking as well as eight effective teaching practices. These findings support the idea that video can be used to enhance teachers' professional learning of the MTPs and the noticing of student thinking (Jacob et al., 2010; Sherin and van Es, 2009; van Es and Sherin, 2002, 2008). The online video-based LS design has the potential to support collaboration of teachers from varying contexts, promote professional growth, and demonstrate how educators might leverage peer coaches within and across schools to develop teachers along the professional continuum. To support teachers' abilities to enact ambitious instruction, we embedded our video-based peer coaching in a PD model, engaging a collaborative team of teachers in cycles of structured observation and reflection supported by video. Throughout the PD, teachers also engaged with peers in professional support groups as they enacted ambitious teaching that involved vetting rich tasks, video recording their teaching enactment, and co-reflecting on practice in several mathematics lessons. One of the affordances of video-based LS with peer coaching includes publicly sharing teachers' practices and supporting one another in their continuous improvement cycle.

Teachers who engaged in this LS opened up their classrooms and their teaching practices, which can be intimidating and has a level of vulnerability that comes with asking for feedback, through the use of video. Moreover, the collegial exchanges on the video annotation tool allowed for peers to push one another toward ambitious teaching while validating, empathizing, encouraging, and pressing one another to go beyond their comfort zone to enact ambitious practices. Each teacher took on the role of being the focal teacher as well as a peer coach, working on their professional noticing at two levels. As a focal teacher, they self-reflected as they attended, interpreted, and responded to the classroom video that they shared with their video LS groups. As a peer coach, they shared their professional noticing as they attended, interpreted, and responded to how the focal teacher enacted the teaching practices in the video. In addition, teachers instantiated their own practice by marking instances of enacting the teaching practice in their own classroom sharing their “glows and grows” through their annotation and self-reflection as well as commenting as peer coaches on these instantiations as they validated and encouraged their peers. Through these professional activities, we believe teachers supported one another to build their “pedagogical courage,” which we used to refer to the teachers' willingness to open up one's practice while being brave to take on the challenge of enacting ambitious teaching practices and refining one's own pedagogy while collectively reflecting with other peers (see Figure 3).

This study shares a “use case” for using video LS with peer coaching to support teachers' enactment of ambitious practices in situ and development of their craft and confidence while unpacking the enactment of the teaching practices using a rich task. We found that the PD design and the specific video LS with peer coaching allowed the teachers to focus on important elements of planning, implementing and reflecting on student learning. Having a video LS team allowed for the articulation and support needed to push the boundaries of the teaching practices with which the teachers were comfortable.

Limitations

There were two primary limitations within this study: (1) not all the LS groups were made up of school-based teams and (2) the five to seven minute video clips did not provide access to the entire enacted lessons. Although we worked to recruit school-based teams, not all LS groups were made up of teachers from the same school, as is suggested by the LS research (Takahashi and McDougal, 2016). Due to the nature of members being at different school sites created the impetus to use an online platform (i.e., GoReact) to collaboratively reflect on the videos. In the end the teachers were able to develop their relationships during the face-to-face PD and engage in the virtual discussions thereafter, which has important implications (discussed below) for future work with video-based lesson study. A second limitation of this study is the length of the video clips that we asked teachers to upload and reflect on. Although this is a limitation because it did not allow the LS groups to reflect on one entire lesson, it had the benefit of requiring the focal teacher to review their video for the clip they wanted to choose. It also allowed the focal teacher to look for a clip that was purposely selected to highlight their research purpose (i.e. the MTP they had set as their personal goal), which focused the LS group discussions on that practice.

Implications

One silver lining to the disruption caused by COVID-19 was how educators were quickly able to mobilize and teach and learn about innovative ways to engage learners in online environments. Likewise, the affordances of online collaboration and video annotation tools offer another way for mathematics teacher educators and professional development providers to consider providing professional learning to teachers. This LS model allowed time and space for peers to build relationships during the face-to-face PD, which may have facilitated the pedagogical courage that the teachers exhibited when showcasing their teaching practices through video. This model also allowed peer coaches to support one another while pushing the boundaries of their teaching repertoire. Using video-based LS is a way to bring support to teachers in small and/or rural areas who might not have a team of teachers within their own building with whom to plan and reflect. Moreover, video-based LS may help to overcome barriers to the implementation of LS, such as the challenge of scheduling a common release time for lesson observation and the financial burden of funding substitute teachers for release time. Another barrier to LS is the lack of access to a knowledgeable other. However, the implementation of video-based LS allows for the knowledgeable other to view videos and provide feedback for more teachers without geographic boundaries. This model has the potential to be scaled up. Finally, another implication for video-based lesson study is the opportunity to capture thoughtful teaching moments through video that provide a picture of practice for other educators to benefit from watching. One of the exciting contributions of LS is the dissemination, publication, and distribution of the public research lessons to a wider audience. Through video-based LS, not only do we have the publishable research lesson to disseminate but we also have a digital record of the lesson captured in the video vignettes. These videos with research lessons could be used for professional learning purposes for the wider LS community.

Conclusion

The most beneficial feature of the video LS is that the video allows all observers, the focal teacher who delivered the lesson as well as the peers who are serving as coaches, to have a chance to “slow down” the teaching to really focus on the research goal that they had chosen to improve upon. We found that teachers were reflective of the instantiation of their own practice with both positive comments as well as self-critique around three of the most demanding practices, using and connecting representations, posing purposeful questions, and supporting productive struggle in learning mathematics. These three practices require teachers to attend, interpret and respond in situ, during the moment in the lesson when this happens with students, thus requiring adaptive and responsive teaching. By collectively experiencing these teaching episodes via video, teachers had a chance to see one's self and how other colleagues experienced similar challenges with different students and this commonality seemed to bring about a collegial understanding and support network. As part of Cobb et al.'s (2018) work in supporting teachers' development of ambitious and equitable MTPs in the Middle School Mathematics and the Institutional Setting of Teaching project, they found that seeing evidence that other teachers' students encountered mathematics challenges helped them to be less defensive about the quality of their instruction and move toward actionable solutions for their next lessons. When teachers reflect on their students' learning and their practices together, it demonstrates that teaching challenges are normal while also providing emotional support to persevere despite the uncertainty that comes with innovation (Cobb et al., 2018). Similarly, we found that teachers who served as peer coaches commented on student thinking and echoed when they witnessed the same challenges that they faced in their own classrooms of responding to students in the moment with purposeful questions and supporting productive struggle. They validated common challenges and provided many more compliments to support their peers' ambitious instruction. Through these findings, we offer one adaptation of a model of video-based Lesson Study using a video annotation tool to instantiate moments of teachers enacting mathematical teaching practices and to leverage peers as coaches to support, press on and validate teachers' efforts in enhancing their pedagogy. We hope this model may serve as a strategy for the mathematics education community to use to learn professionally in these uncertain times and offer yet another powerful adaptation of Lesson Study to serve the LS community at large.

Figures

Figure 1

Adapting lesson study with video peer coaching

Figure 2

Video-based LS with peer coaching

Figure 3

Supporting “pedagogical courage” to refine one's practice

Table 1

Eight high-leverage mathematics teaching practices (NCTM, 2014)

MTP 1. Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions

MTP 2. Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies

MTP 3. Use and connect mathematical representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving

MTP 4. Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments

MTP 5. Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance students' reasoning and sense making about important mathematical ideas and relationships

MTP 6. Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems

MTP 7. Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships

MTP 8. Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning

Table 2

Teachers' noticings about their own enactment of the MTPs

Teacher noticings about their enactment of the MTPs	Positive instances	Self-critique	Total
MTP 1. Establish mathematics goals to focus learning	1	2	3
MTP 2. Implement tasks that promote reasoning and problem solving	10	2	12
MTP 3. Use and connect mathematical representations	13	15	28
MTP 4. Facilitate meaningful mathematical discourse	8	7	15
MTP 5. Pose purposeful questions	7	15	21
MTP 6. Build procedural fluency from conceptual understanding	3	4	7
MTP 7. Support productive struggle in learning mathematics	7	12	19
MTP 8. Elicit and use evidence of student thinking	4	9	13
Total	52	65	117

Table 3

Patterns of interaction among peer coaches and focal teachers across the nine lesson study groups

Patterns of interaction among peer coaches and focal teachers	Frequency
Complimenting a teacher's move	73
Pressing teachers toward more ambitious teaching	37
Validating a common challenge	15
Connecting to student thinking	80

Table 4

Example lesson study group: transcript of video clip with focal teacher and peer exchanges

Excerpt of classroom video episode	Focal teacher and peer annotation exchanges on the video vignette	Patterns of interaction instantiation of MTP
Student: So I switched. My original total was this. (points to unbalanced team). This was not fair at all. (pointing) This was not either T: Which team would you rather be on? He gestures… in this one really in the slow team? Student: Oh Never mind. (laughter)	Vanessa (Focal teacher): I think this part of the video showcases what happens when you elicit student thinking. Looking at this paper myself I probably would not have been able to decipher why he subtracted 3.07 or that he had a very methodical reasoning as to why he wanted to switch his team mates to make them better	Positive Instantiation, MTP 8: Vanessa highlights a positive instantiation in her own practice as she elicited student thinking. She notes that this eliciting helped her to understand what the student was thinking more so than just looking at his paper would have
T: Okay remember okay okay what about up here. What did you do? Student: So I switched 8.7 and 11.46 because I wanted to switch their lowest number with this one . So we can make this team like it faster and more balanced so then I went on the back and because it's 11 minus 8 equals 3 so I did minus 3 because there's no point for subtracting 11 just so I did that got 3.46 and I got 40.73 and it's like off by a second Teacher: Very very close (to another group working) What were you gonna say ladies? Student: We got each one to the same beginning decimal … NEAT!. because before I rounded each decimal to the closest and try to sort each team by them so I can add them and see which one's the closest and so the 42 was the lowest one but this one was originally at 36 and this one was a 46 and so I kept changing I kept drawing a different number each time from each team. I've kept swapping … swapping members from each team until I got the closest number and range which is 42.	Vanessa (Focal teacher): At this moment I wish I would have had better questioning. I must admit, I did not believe her and I couldn't find her mistake quickly. She did in fact make a calculation error and team 1's total should really be 41.28, not 42.28. I wish I would have asked about her initial rounding method. I could have even brought in more student to student discourse and have her compare answers with her classmate. I'm disappointed that I just settled for her response	Self-critique, MTP 5, 8, 4: Vanessa not only engages in self-reflection of her practice but also considers varying strategies of how to improve her practice. Possible strategies include eliciting student thinking as well as promote discourse amongst the students. She demonstrates her vulnerability by describing her disappointment
	Dana (Peer coach): I think this is an amazing example of “implementing a task that promotes reasoning and problem solving”. It was so interesting to hear the students explain how they tried to make the teams even, especially when they had to try a second or third strategy. Their reasoning was really insightful into how they saw the problem, and even to determine which team they would want to be on	Complimenting a teacher's move: Peer validates the teacher's work on a completely different MTP, promoting reasoning and problem solving, encouraging her peer. Connecting to student thinking: Peer focused in on student thinking and how this connected to the MTP. Being able to solve a task in multiple ways is a hallmark of a high cognitive demand task
	Dana (Peer coach): I think it would have been interesting to hear students discuss their strategies with each other and talk about how they decided to put their teams together and what worked and what did not	Pressing teachers toward more ambitious teaching: Peer encourages Vanessa to grow in the area of facilitating discourse which she already identified in her self-reflection
	Katie (Knowledgeable other): This was such a fun video to watch. The interaction between you and the students is so wonderful. Your students did a great job explaining their thinking to you, but I echo Dana's comment of wishing I saw the part where they talked with one another about the problem. You can tell the students were engaged in this task	Complimenting a teacher's move: The knowledgeable other gives a glow for eliciting student thinking. Pressing teachers toward more ambitious teaching: She also presses the focal teacher in facilitating discourse, validating the teacher and the other peer coach at the same time. Connecting to student thinking: The knowledgeable other made a direct observation of student engagement.

Table A1

Theme, description and examples of peer exchanges during video lesson study

Theme	Description	Example
Complimenting a teacher's move	Peer coaches marked instances of the teacher enacting teaching practice and complemented the focal teacher's move as in this annotation	“Throughout the lesson you were observing the different ways students understood the task and you were encouraging mathematical discourse. You rephrased and gave on-the-go feedback, not negatively at all, so students seemed (even with low voices) motivated and encouraged to share their thoughts, drawings, and graphs. You gave them this vocabulary and options before starting the task. I appreciate how you led the lesson and how you calmly empowered your students to participate.”
Pressing teachers toward more ambitious teaching	There were two ways peer coaches approached pushing teachers towards ambitious teaching. Most often, they would suggest or state, “I wished that I saw more xyz.” Another way of pushing them was by asking questions for the teacher to consider. At times, the peer exchange revealed teachers working through a common dilemma of practice	“I encourage you as you reflect on your lesson, to think about the types of purposeful questions you are going to ask and find a way to quickly refer to them in your revised lesson. One way that helped me was posting them around the room as a quick reference. Another strategy is to have them on a clipboard so that you have them as you move around. Another thought may be to provide question stems at each of the groups.”
Validated a common challenge	Peer coaches and teacher exchanges validated a common challenge and encouraged the teacher who might have voiced difficulty in enacting a practice	“Don't beat yourself up so much! I think you did a great job. It is hard to try to manage all the different questions and tasks we want to do and snapshot it in a short video. I'm sure you did everything you intended to at some point in the whole lesson. I love that the way your students approached this lesson is a lot different than the way my students approached it but all of our students got the same critical thinking and rich task piece out of it.”
Connecting to student thinking	Other comments focused on the strengths peer coaches saw in the students in the video	One peer commented “I liked how the student used ratios or proportions and their own height and foot measurement. ‘and immediately in the next sentence, attributes it to the teacher's action,’ How you Ann posed the questioning encouraged students to volunteer and expand their thinking.”

Appendix 1 Crab walk relay

Crab walk relay race

Mr. Fair, the physical education teacher, is forming three crab walk relay teams. There must be four students on each relay team. To determine the teams, Mr. Fair uses the students' crab walk times from the last physical education class. The students are timed to the nearest tenth of a second.

Student time in seconds

Harry 12.2	Ryan 9.91	Brooke 10.9
Sally 10.3	Josh 11.46	Travis 11.11
Joseph 8.79	Meredith 10.54	Greg 11.9
Amy 8.7	Anna 9.19	Emily 11.1

Mr. Fair wants each of the three relay teams to be as equally matched in total time as possible. Mr. Fair writes the three teams on a piece of paper. What four students could Mr. Fair put on each team? Mr. Fair decides to write the less than, greater than, or equal symbols on his paper to compare the three teams. What could be on Mr. Fair's paper? Answer the question using pictures, words, tables, graphs and/or symbols.

Appendix 2